Where do the statistical models come from and what are their exact equations?

FastBMD currently offers 10 models as shown below

These statistical models are the same models the the US Environmental Protection Agency and the National Toxicology Program have used in their studies on 'omics dose-response analysis. They are the same models available in BMDExpress, and are the ones recommended in the peer-reviewed report that outlines the National Toxicology Program Approach to Genomic Dose-Response Modeling. Exp2 - Exp4 are four different forms of exponential models. Poly2, Poly3, and Poly4 are polynomial models with degree 2, 3, and 4. More details and the full mathematical forms can be found in the NTP report linked above.